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SCHOOL    DOCUMENT    NO.   ^-1918 

BOSTON  PUBLIC  SCHOOLS 

ARITHMETIC 

DETERMINING   THE    ACHIEVEMENT    OF    PUPILS    IN 
COMMON    FRACTIONS 


Bulletin  No.  XV.  of  the  Department  of 
Educational   Investigation    and    Measurement 


jiJlsiR>,  ;i©lB';\;' ',;;',,'  >',^:;  ;  ;;'. 


BOSTON 

PRINTING     DEPARTMENT 

IQ18 


In  School  Committee,  Boston,  June  26,  1918. 
Ordered,  That  four  thousand  (4,000)  copies  of  a 
bulletin  on  ''Determining  the  Achievement  of  Pupils 
in  Common  Fractions,"  prepared  by  the  Department 
of  Educational  Investigation  and  Measurement  and 
approved  for  publication  by  the  Board  of  Superin- 
tendents at  its  meeting  on  June  19,  1918,  be  printed 
as  a  school  document. 

Attest : 

THORNTON   D.   APOLLONIO, 

Secretary. 

I 


,'  -ij. 


/ 


CONTENTS, 


Introduction    .        .        .        .        •        •        •        • 
I.     Previous  Work  in  Fractions  in  Boston 

Summary  of  Results  in  Addition  of  Fractions, 
Summary  of  Results  in  Subtraction  of  Frac- 
tions   

II.     Multiplication  and  Division  of  Fractions 

Extent  of  Tests 

Types  of  Fractions      .... 
Construction  of  Tests 
Giving  Tests  and  Correction  of  Results 
III.    Analysis  of  Results  .... 

Achievement 

Analysis  of  Results  in  Grades  VII  and  VIII 
Diagnosis  of  Results  in  Each  Test 
Test  1 
Test  2 
Test  3 
Test  4 
Tests 
Teste 
Testy 
Analysis  of  Results  in  Grade  VI 
IV.     Plan  of  Diagnosis  for  Teacher 
V.     Summary  and  Conclusions     . 


Page 

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387243 


Digitized  by  the  Internet  Archive 

in  2007  with  funding  from 

IVIicrosoft  Corporation 


http://www.archive.org/details/arithmeticdetermOObostrich 


INTRODUCTION. 


As  fast  as  practicable  the  Department  of  Educational 
Investigation  and  Measurement  has  undertaken  to 
extend  educational  measurement  in  arithmetic  beyond 
the  four  fundamental  operations  covered  by  the  Courtis 
Tests.  To  that  end  from  time  to  time  the  department 
has  given  tests  in  the  four  processes  of  common  fractions, 
and  in  a  more  limited  way  in  problem  work.  This 
bulletin,  prepared  by  the  Assistant  Director,  Mr.  Arthur 
W.  Kallom,  covers  the  work  done  thus  far  with  the 
addition,  subtraction,  multiplication,  and  division  of 
common  fractions.  Particular  attention  is  directed  to 
Mr.  Kallom's  discussion  of  the  various  types  of  problems 
involved  in  computation  with  fractions,  and  also  to  the 
suggestions  offered  of  ways  and  means  of  improving 
unsatisfactory  achievements  of  pupils. 

It  is  frequently  argued  in  educational  discussions  that 
command  over  the  tools  of  an  education  should  be 
acquired  by  the  end  of  the  sixth  grade.  Obviously, 
ability  to  compute  with  common  fractions  is  merely  a 
means  to  a  desired  end  and  not  an  education  in  itself. 
This  study  shows  that  either  (a)  not  as  much  ability 
to  use  common  fractions  is  being  developed  before 
the  end  of  the  sixth  grade  as  should  be,  or  (b)  the  period 
of  the  first  six  years  in  the  elementary  school  is  too 
short  to  furnish  pupils  with  all  the  tools  of  an  education. 

The  manuscript  for  this  bulletin  was  approved  for 
publication  by  the  Board  of  Superintendents  at  its 
meeting  on  June  19,  1918. 

FRANK  W.   BALLOU, 

Assistant  Superintendent  in  Charge. 


.^..rrSCHOOL  DOCUMENT   NO.    5. 


DETERMINING  THE  ACHIEVEMENT  OP  PUPILS  IN 
COMMON  FRACTIONS. 


I.     Results  of  Previous  Tests  in  Fractions. 

Tests  in  the  four  fundamental  operations  have  been 
given  in  the  City  of  Boston  during  the  past  five  years. 
During  this  time  the  gain  in  the  amount  of  work  done  has 
been  from  12  per  cent  to  17  per  cent.  This  gain  in 
amount  of  work  done  has  been  accompanied  by  an  actual 
increase  in  the  accuracy  with  which  the  work  was  com- 
pleted. This  increase  by  which  pupils  are  being  graduated 
or  are  being  promoted  into  the  next  grade  with  varying 
degrees  of  superiority  up  to  17.7  per  cent  over  the 
results  which  were  being  obtained  previous  to  the  giving 
of  the  Courtis  standard  tests,  is  due  directly  or  indirectly 
to  the  system  of  educational  measurement  as  established 
in  Boston. 

The  ability  to  handle  integers  does  not  constitute, 
however,  the  sum  total  of  the  tools  necessary  for  the 
child  in  order  that  he  may  do  arithmetic.  Fractions,  in 
one  form  or  another,  play  a  large  part  in  the  arithmetical 
work  of  the  pupil.  That  we  might  know  how  well  the 
pupils  are  doing  their  work  in  common  fractions,  a  plan 
was  organized  in  1915  to  give  tests  in  addition,  subtrac- 
tion, multiphcation,  and  division  of  fractions  in  successive 
years  to  a  group  of  approximately  1,000  children  in 
Grades  VI,  VII,  and  VIII  in  an  experimental  way. 
The  tests  were  organized  by  the  department  in  such  a 
way  as  to  determine  not  only  what  the  ability  was  to  do 
the  various  operations,  but  also  if  the  pupils  failed,  in 
what  type  of  examples  in  any  given  operation  the  pupil 
failed.  Of  the  two  phases  of  the  work,  the  latter  is  of 
the  greater  importance.  It  is  not  enough  to  say  pupils 
fail  to   do   addition   of  fractions  with   a   speed   or   an 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.      7 

accuracy  which  seems  desirable.  One  must  go  further 
and  show  the  types  of  examples  in  which  pupils  fail. 

Further,  if  a  single  test  be  given  in  a  certain  operation 
and  a  pupil  fails,  it  becomes  the  work  of  the  teacher  to 
determine  in  what  particular  the  pupil  fails.  This 
enables  the  teacher  to  place  the  emphasis  upon  the  work 
where  it  belongs  and  not  waste  her  efforts  and  those  of 
the  pupils  in  drilling  on  matter  which  needs  no  drilling. 

Addition  of  fractions  may  be  divided  into  fourteen 
types;  *  subtraction  of  fractions  may  be  divided  into 
nine  types  similar  to  those  in  addition,  the  difference 
in  number  being  due  to  the  fact  that  it  is  impossible  to 
reduce  any  answer  in  subtraction  to  a  mixed  number. 
The  types  in  multiplication  and  division  will  be  analyzed 
in  the  succeeding  pages  of  this  bulletin. 

Summary  of  Results  in  Addition  of  Fractions. 
The  results  of  the  tests  in  addition  of  fractions  were 
published  in  School  Document  No.  3,  1916.  The  data 
upon  which  the  conclusions  were  drawn  are  shown  in  the 
following  tables.  Table  I  shows  the  type  of  examples 
used  in  the  six  tests  together  with  the  time  allowance. 

TABLE   L 
Showing    Examples    Used    in    Tests    in    Addition    of    Fractions, 


December,  1915. 

Addition 

of 

Fractions.— 

-Test  1.— Time, 

2  Minutes. 

m  \ 

(2)     f^ 

(3)   fe 

(^)   To 

1 

_4 

1 

14 

7 

11 

7 
i2 

Addition 

of 

Fractions.— 

-  Test  2.—  Time, 

2  Minutes. 

™i 

<.,  1 

(«  i 

«  1 

1 
_6 

3 

1 

\2 

7 

*  See  School  Document  No.  3,  1916.     "Determining  the  Achievement  of  Pupils  in  Addi- 
tion of  Fractions." 


8 


SCHOOL   DOCUMENT   NO.  5. 


Addition  of  Fractions. —  Test  3. —  Time,  2  Minutes, 

I  (2)      I  (3)       I  (4)     g 

II  1  U  2 
15                     2                       14                       3 


(1) 


Addition  of  Fractions- 

-  Test  4. 

(1)     \             (2)      1 

(3) 

9  1 

10  4 

Time, 


Minutes. 

4 

9 

5 

8 


(4) 


Addition  of  Fractions. —  Test  6. —  Time^ 

1  5^  3 

6  12  8 


(^)  ii) 


(2) 


Minutes. 
12 
10 


(4) 


Addition  of  Fractions. —  Test  6.- —  Time,  2  Minutes. 


9^ 
10 


(2) 


(3) 


1 

8 
9^ 
10 


(4) 


12 
]_ 
10 


Table  II  shows  the  medians  obtained  as  a  result  of 


the  tests. 


TABLE   IL 
Summary  Sheet  —  City  Medians. 

Addition  of  Fractions,  December,  1915. 


3% 
1^ 

Test  1. 

Test  2. 

Test  3. 

Test  4. 

Test  6. 

Test  6. 

Grade. 

.2 

1^ 

< 

i 

1^ 

< 

11 

< 

J 

-el's 

is 

d 

VIII 

VII 

VI 

1,130 
1,243 
1,265 

20.7 
16.6 
10.7 

88.0 
87.0 
80.0 

11.6 
10.1 

7.7 

74.0 
73.0 
66.0 

8.4 
7.3 
6.5 

47.0 
46.0 
42.0 

6.0 
6.3 
4.0 

68.0 
69.0 
70.0 

6,9 
6.3 
4.6 

62.0 
65.0 
61.0 

6.4 
5.7 
4.4 

47.0 
48.0 
49.0 

ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.     9 

The  following  conclusions  were  drawn  as  a  result  of  the 
analysis  of  the  tests.* 

1.  The  factors  that  enter  into  the  problem  of  adding 
fractions  are  much  more  complex  than  those  that  enter 
into  the  problem  of  adding  integers. 

2.  The  errors  were  largely  due  to  the  failure  of  pupils 
to  reduce  consistently  either  to  lowest  terms  or  to  mixed 
numbers.  This  faihng  on  the  part  of  many  children 
to  use  the  principle  of  reduction  would  seem  to  indicate 
that  the  method,  now  largely  in  use,  of  teaching  such 
reductions  by  themselves,  has  failed  to  produce  satis- 
factory results.  In  view  of  this  fact,  would  it  not  be 
well  to  teach  reductions  as  such,  in  connection  with  the 
subject  of  addition  of  fractions?  This  would  at  least 
make  a  closer  connection  between  the  two  operations, 
and  thereby  tend  to  form  the  habit  of  writing  the  answer 
in  its  best  form. 

3.  Eight  per  cent  of  the  pupils  in  Grade  VI,  11  per 
cent  in  Grade  VII,  and  5  per  cent  in  Grade  VIII  were 
unable  to  do  the  simplest  problems  in  the  addition  of  * 
fractions. 

4.  Drill  and  individual  work  given  the  children  in 
Grade  V  of  selected  schools  in  the  spring  at  the  sug- 
gestion of  the  department  showed  its  effect  in  the  work 
of  Grade  VI  in  the  late  fall.  This  was  evidenced  by  an 
increase  in  both  speed  and  accuracy  over  that  obtained 
in  the  entire  city  and  in  two  cases  over  that  shown  by 
the  whole  number  of  pupils  in  the  grade  in  which  the 
selected  groups  were  enrolled. 

Summary  of  Results  in  Suhtr action  of  Fractions. 
The  results  of  the  tests  in  subtraction  of  fractions 
were  not  published  because  they  were  not  materially 
different  from  the  results,  of  the  tests  in  addition.  The 
following  table  shows  the  types  of  examples  used  in  the 
five  tests  together  with  the  time  allowance. 

*  School  Document  No.  3. 


10  SCHOOL   DOCUMENT  NO.  5. 

TABLE   III. 

Showing  Examples   Used   in  Tests  in   Subtraction   of  Fractions, 

December,  1916. 

Subtraction  of  Fractions. —  Test  1. —  Time,  2  Minutes. 
(1)     \  (2)     I  (3)     I  (4)     f^ 

1  1  1  A 

_4  _4  JS  16 

Subtraction  of  Fractions. —  Test  2. —  Time 
(1)     \  (2)     I  (3) 

1  3 

Subtraction  of  Fractions. —  Test  3. 
(1)     I  (2)     I  (3) 

1  _3 

i2  i2  —  _ 

Subtraction  of  Fractions. —  Test  4- —  Time,  2  Minutes. 
(1)     4.  (2)     6  (3)     6  (4)     6 

2i  5|  2f  ^ 

Subtraction  of  Fractions. —  Test  5. —  Time,  2  Minutes. 
(1)     9i  (2)     7A        (3)     7,V        (4)     74 

iJ  6f  4f  2fv 

In  the  tests  in  addition  the  addition  of  mixed  numbers 
was  not  included  although  it  is  very  probable  that  there 
would  have  been  some  difficulty  in  disposing  of  the  sum 
of  the  fractions,  especially  if  the  sum  were  more  than 
an  integer.  This  same  phase  occurs  in  the  multiplication 
of  mixed  numbers  by  an  integer  and  will  be  pointed  out 
in  its  proper  place.  The  subtraction  of  mixed  numbers, 
however,  is  a  vital  problem  especially  when  the  fraction 
in  the  subtrahend  is  larger  than  the  fraction  in  the 
minuend.  Because  of  this.  Tests  4  and  5  were  given 
upon  this  type  of  example.  Table  IV  shows  the  medians 
in  speed  and  accuracy  in  the  subtraction  of  fractions. 


-  Time, 

2  Minutes. 

2 
3 

(4) 

3 

4 

3 
11 

5 

9 

-  Time, 

2  Minutes. 

7 
9 

(4) 

7 
10 

1 
12 

8 
15 

ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.    U 


TABLE   IV. 
Summary  Sheet  —  City  Medians. 

Subtraction  of  Fractions,  Deoember,  1916. 


Pupils. 

Test  1. 

Test  2. 

Test  3. 

Test  4. 

Test  5. 

Grade. 

1^ 

f 

II 
1^ 

II 

d 

|l 

d 
.2 

VIII 

1,239 
1,283 
1,499 

22.5 
19.7 
15.1 

91.0 
84.0 
73.0 

7.3 
6.0 
4.9 

86.0 
85.0 
76.0 

6.1 
5.6 
4.6 

65.0 
61.0 
51.0 

18.0 
14.2 
11.9 

99.0 
97.0 
85.0 

6.4 
5.2 
4.6 

81.0 

VII 

66.0 

VI 

64.0 

The  following  summary  shows  the  number  of  districts, 
the  number  of  grade  classes,  the  grades  tested,  and  the 
total  number  of  pupils  included  in  the  test  in  addition 
and  subtraction  of  fractions. 

Addition  of  Fractions. 
December,  1915. 

Number  of  elementary  districts 12 

Number  of  grade  classes 91 

Given  in  grades VIII,  VII,  VI 

Number  of  pupils 3,638 

Subtraction  of  Fractions. 

December,  1916. 

Number  of  elementary  districts       ...        .        .        .  10 

Number  of  grade  classes 102 

Given  in  grades VIII,  VII,  VI 

Number  of  pupils ,        4,021 

II.     Multiplication    and    Division    of    Fractions. 

Extent  of  Tests. 
The  following  summary  shows  the  number  of  districts, 
the  number  of  grade  classes,  the  grades  tested,  and  the 
number  of  pupils  included  in  the  test  in  multiplication 
and  division  of  fractions,  given  in  December,  1917. 

Number  of  elementary  districts       .....  10 

Number  of  grade  classes 95 

Given  in  grades VIII,  VII,  VI 

Number  of  pupils 3,513 


12  SCHOOL  DOCUMENT  NO.  5. 

Types  of  Fractions. 

An  extended  analysis  was  made  in  addition  and  sub- 
traction of  fractions  to  determine  the  various  types 
with  which  the  pupil  came  in  contact.  An  analysis  on 
a  similar  basis  of  the  processes  of  multiplication  and 
division  of  fractions  was  not  believed  necessary.  In 
these  two  processes  the  separation  into  types  depended 
upon  the  character  of  the  multiplier  and  multiplicand 
or  the  dividend  and  divisor. 

The  process  of  division  in  fractions  is  either  one  of  two 
procedures.  In  one  case  one  proceeds  to  find  how  many 
times  one  number  of  a  certain  denomination  is  contained 
in  a  larger  number  of  the  same  denomination.  This  is  to 
determine  how  many  measures  of  a  certain  length  there 
are  in  a  measure  of  a  different  length.  This  type  of 
procedure  has  beea  called  division  hy  measuring.  In  the 
other  case  one  proceeds  to  separate  a  number  into  a 
certain  number  of  parts.  This  type  of  division  is  called 
division  hy  parting.  These  two  types,  measuring  and 
parting,  became  the  basis  upon  which  the  three  tests  in 
division  of  fractions  were  formulated.  In  view  of  these 
conditions  the  following  types  were  selected: 

Multiplication. 
Integer  multiplied  by  fraction. 
Fraction  multiplied  by  integer. 
Mixed  number  multiplied  by  integer. 
Integer  multiplied  by  mixed  number. 
Mixed  number  multiplied  by  fraction. 
Fraction  multiplied  by  mixed  number. 
Mixed  number  multiplied  by  mixed  number. 
Fraction  multiplied  by  fraction. 

Division. 
Integer  divided  by  fraction  (measuring) . 
Fraction  divided  by  integer  (parting) . 
Mixed  number  divided  by  integer  (parting) . 
Integer  divided  by  mixed  number  (measuring). 
Fraction  divided  by  fraction  (measuring). 
Mixed  number  divided  by  fraction  (measuring). 


n 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.   13 

The  two  types,  fraction  divided  by  mixed  number  and 
mixed  number  divided  by  mixed  number,  are  not 
included  because  they  do  not  conform  to  either  the 
parting  or  measuring  criterion.  In  practical  work  we 
neither  have  to  perform  such  examples  as  dividing  a 
fraction  into  3|  parts  nor  finding  how  many  3|  inches 
there  are  in  \  of  an  inch.  Neither  are  we  required  to 
perform  such  examples  as  dividing  a  mixed  number  into 
3 1  parts  nor  finding  how  many  3|  inches  there  are  in 
4i  inches. 

Of  course,  it  is  recognized  that  common  fractions  are 
taking  less  and  less  place  in  our  practical  life,  the  proc- 
ess giving  way  more  and  more  to  the  use  of  the  decimal 
fraction.  However,  there  is  still  use  for  the  common 
fraction  having  a  small  denominator,  and  it  is  still  a 
part  of  the  required  work  in  our  courses  of  study.  This 
being  true,  it  is  pertinent  to  ascertain  what  results  we 
are  achieving. 

Construction  of  the  Tests. 
In  constructing  the  tests  the  department  decided 
in  the  light  of  previous  experience  with  addition  and 
subtraction  of  fractions  that  multiplication  and  division 
might  be  given  at  one  time.  In  order  to  decrease  the 
number  of  tests,  two  types  were  placed  in  a  test.  For 
example,  multiplication  of  an  integer  by  a  fraction  and 
multiplication  of  a  fraction  by  an  integer  comprised 
Test  1.  As  will  be  seen  in  Table  V,  the  other  tests 
were  made  in  a  similar  way.  In  the  analysis  of  the 
results  the  two  types  will  be  discussed  separately. 
There  was  an  effort  to  keep  the  tests  within  the  realm 
of  the  practical.  In  all  cases  the  terms  of  the  fractions 
involved  were  kept  small.  A  fraction  multiplied  by  a 
fraction  is  not  in  any  test  but  is  included  in  the  process 
of  multiplication  of  mixed  number  by  a  fraction.  The 
latter  type  was  used  because  it  was  considered  more 
difficult.  If  this  be  true,  a  pupil  might  be  able  to  do 
the  former  type  but  unable  to  do  the  latter.  However, 
ability  to  do  the  latter  would  include  abihty  to  do  the 


14  SCHOOL  DOCUMENT    NO.  5. 

former.     Table  V  shows  the  types  of  examples  and  the 
time  allowance  for  each  test. 


TABLE  V. 

Showing  Examples  Used  in  Tests  in  Multiplication  and  Division  of 
Fractions,  December,  1917. 

Multiplication  of  Fractions. —  Test  1. —  Time,  2  Minutes. 
(1)     I  X  6      (2)     i  X  8       (3)     f  X  12      (4)     12  X  A 

Multiplication  of  Fractions. —  Test  2. —  Time,  4  Minutes, 

(1)     2461     (2)     5731-     (3)     275       (4)     456|     (5)     189 
5  5  8|  2  5i 


Multiplication  of  Fractions. —  Test  3. —  Time,  2  Minutes. 
(1)     4|  X  i      (2)     7i  X  I      (3)     5^  X  I      (4)     |-  X  2f 

Multiplication  of  Fractions. —  Test  4- —  Time^  5  Minutes. 

(1)       321     (2)       84J     (3)       29i         (4)     25i     (5)     191 
69  J  791  28i  17f  m 


Division  of  Fractions. —  Test  5. —  Time,  2  Minutes. 
(1)     i  -^  8        (2)     9  -^  f        (3)     6  +  i        (4)     8  -^  t 

Division  of  Fractions. —  Test  6. —  Time,  4  Minutes. 
(1)     5678J  ^  5         (2)     27891  -^  4        (3)     2467  ^  8i 

(4)     6752  ^  12| 

Division  of  Fractions. —  Test  7. — ■  Time,  3  Minutes. 
(1)     f-4      (2)     3f  ^i      (3)     5t-f      (4)     6f-^f 

Giving  of  the  Tests  and  Correction  of  Results. 

Following  the  plan  developed  in  1912  and  continued 

since    the    department     was     organized,*    twenty-five 

Normal  School  seniors  were  trained  to  give  the  tests 

in  a  uniform  manner.     The  tests  were  given  to  1,290 

*  Ballou,  F.  W.,  "Training  Normal  School  Seniors  in  Educational  Measurement,"  School 
and  Society,  Volume  V.,  No.  108,  pages  61-70,  January  20,  1917. 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.    15 

pupils  in  Grade  VI,  1,196  pupils  in  Grade  VII,  and  to 
.  1,027  pupils  in  Grade  VIII  in  December,  1917. 

The  old  course  of  study  for  Grade  V  requires  : 

Multiplication  of  fractions  and  mixed  numbers  and 
integers;  finding  fractional  parts  of  integers  includ- 
ing the  cases  where  the  parts  so  obtained  are  mixed 
numbers. 

Thus  the  sixth  grade  may  begin  their  work  without 
a  knowledge  of  division  of  fractions,  and  it  is  possible 
that  division  of  fractions  may  not  have  been  taught 
during  the  first  three  months  of  the  school  year.  In 
spite  of  this  knowledge,  it  was  decided  to  test  in  Grade 
VI  for  two  reasons.  First,  that  it  might  be  known  just 
what  the  status  of  Grade  VI  actually  is  on  a  city- 
wide  basis  in  multiplication  and  division  of  fractions; 
and  second,  to  find  out  what  is  done  by  those  schools 
which  did  more  work  than  was  actually  required  by 
the  course  of  study. 

After  completing  the  work,  the  examiners  brought  the 
tests  to  the  office  of  the  department  and  all  the  work  of 
correction  and  tabulation  was  done  by  members  of  the 
department.  Certain  rules  were  formulated  for  the 
correction  of  results. 

(a)     All  results  which  were  not  reduced  to  lowest  terms  or  to 

mixed  numbers  were  called  wrong. 
(5)     The  papers  on  which  children  added  or  subtracted  the 

fractions  were  counted  as  I.  N.  F.  papers.    (Instructions 

Not  Followed.) 

(c)  All  other  papers,  regardless  of  how  the  child  did  the 

examples,  were  scored  as  right  or  wrong. 

(d)  The  form  of  doing  the  work  did  not  count  against  the  child 

if  his  answer  was  correct. 

(e)  Some  children  did  not  do  Test  1,  but  started  upon  Test  2, 

owing  to  confusion  in  understanding  the  directions. 
Any  paper  showing  no  work  at  all  in  Test  1  was  marked 
I.  N.  F.  in  all  tests.  (Instructions  Not  Followed.) 
These  were  very  few. 


16 


SCHOOL  DOCUMENT   NO.  5. 


(/)  If  in  Tests  5,  6,  and  7  (division  of  fractions)  pupils 
multiplied,  the  papers  were  not  niarked  I.  N.  F. 
(Instructions  Not  Followed.)  This  was  because  there' 
is  much  confusion  between  the  two  processes  and  many 
pupils  really  multiply  when  they  think  they  are  divid- 
ing. If  any  other  process  was  used  the  test  was  marked 
I.  N.  F.     (Instructions  Not  Followed.) 

III.  Analysis  of  Results. 
Achievement. 
Table  VI  shows  the  results  for  the  entire  number 
of  pupils  tested.  In  the  first  column  is  shown  the  grade, 
followed  by  a  column  showing  the  number  of  pupils 
tested  in  each  grade.  Under  each  test  is  given  the  speed 
median  and  the  accuracy  median  for  each  test  and 
grade.  The  table  is  to  be  interpreted  as  follows:  In 
Grade  VIII,  1,027  pupils  were  tested.  These  pupils 
attained  a  speed  median  of  11.1  examples  with  an 
accuracy  median  of  93  per  cent  in  Test  1.  In  Test  2 
the  speed  median  was  8.8  and  the  accuracy  median  was 
63  per  cent.  Thus,  reading  across  the  page  on  the  first 
line  one  will  find  the  medians  in  speed  and  accuracy  for 
each  test  for  Grade  VIII.  The  table  shows  the  same 
facts  for  Grades  VII  and  VI. 

TABLE   VL 
Summary  Sheet  —  City  Medians. 

Multiplication  and  Division  of  Fractions. 


Multiplication. 

Division. 

TEST  1.  . 

TEST  2. 

TEST  3. 

TEST  4. 

TEST  5. 

TEST  6. 

TEST  7. 

Grade. 

i 

CO 

i 

02 

< 

i 

11 

P 

.i 

p 

< 

^1 

13 
03 

1^ 

< 

d 
.2 

03'^ 
< 

vm 

1,027 

11.1 

93 

8.8 

63 

7.6 

85 

4.7 

0 

10.1 

75 

3.3 

29 

10.3 

79 

VII 

1.196 

8.4 

88 

7.7 

38 

6.4 

81 

4.2 

0 

8.2 

59 

2.9 

0 

8.5 

68 

VI 

1,290 

6.2 

13 

8.2 

0 

4.7 

0 

5.6 

0 

5.4 

1 

3.2 

0 

4.9 

0 

ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.   17 

It  will  be  noticed  that  the  accuracy  medians  for 
Test  2,  multiplication  of  mixed  number  by  integer  or 
integer  by  mixed  number,  Test  4,  multiplication  of 
mixed  number  by  a  mixed  number,  and  Test  6,  division 
of  mixed  number  by  an  integer,  are  especially  low. 

It  is  probably  true  that  there  is  no  great  use  for  the 
type  of  work  shown  in  these  three  tests  in  practical  life, 
but  the  business  world  does  require  it  to  some  extent; 
business  courses  in  our  high  schools  require  the  processes, 
and  the  new  course  of  study  requires  this  work.  In 
view  of  these  three  conditions,  it  was  thought  best  to 
include  these  three  tests  in  order  that  we  might  have 
some  facts  on  which  to  base  the  development  of  our 
work  in  multiplication  and  division  of  fractions. 

Analysis  of  Results  in  Grades  VII  and  VIII. 

The  analysis  of  results  which  is  given  in  this  bulletin 
is  based  wholly  upon  a  study  of  the  wrong  examples  in 
the  work  performed  by  pupils  in  the  test  given  in  Decem- 
ber. It  is  perfectly  possible  that  a  pupil  who  does  the 
work  and  reaches  the  right  result  may  be  doing  it  in  an 
inefficient  and  round-about  manner.  When  correcting 
large  numbers  of  papers,  his  work  does  not  attract  the 
attention  that  is  attracted  by  a  pupil  who  does  many 
examples  and  gets  none  or  only  a  few  right. 

This  study  is  based,  then,  upon  those  papers  which 
showed  low  scores  in  accuracy.  Furthermore,  owing  to 
the  low  degree  of  accuracy  in  Grade  VI,  due  largely  to 
lack  of  knowledge,  the  analysis  is  based  upon  work  in 
Grades  VII  and  VIII.  In  a  study  like  the  present  one 
a  piece  of  work  done  by  a  person  ignorant  of  the  process 
has  little  or  no  value.  The  value  of  a  study  of  this  kind 
comes  from  studying  results  of  pupils  who  are  supposed 
to  have  been  taught  the  process.  An  analysis  of  Grade 
VI  will  be  made  in  a  later  part  of  the  bulletin. 

It  is  impossible  for  the  report  of  a  study  to  be  as  helpful 
to  a  teacher  as  if  the  individual  teacher  had  made  the 
study  for  herself.  It  is  only  when  the  teacher  will  take 
the  work  of  her  class  room  and  make  some  similar  analy- 


18  SCHOOL  DOCUMENT  NO.  5. 

sis,  seeking  to  find  out  why  the  pupil  makes  the  failure 
and  just  what  the  pupil  does  in  making  the  failure,  that 
we  are  going  to  make  great  gains  in  the  treatment  of 
individual  pupils.  The  analysis  is  given  here  rather  in 
detail  in  the  hopes  that  it  may  act  as  a  guide  and  stimu- 
late some  teachers  to  undertake  this  rather  laborious 
but  extremely  interesting  work  for  the  good  of  the  indi- 
vidual who  is  having  trouble  with  his  fractions  and  school 
work  in  general. 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  19 


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20  SCHOOL  DOCUMENT  NO.  5. 

Table  VII  shows  the  general  situation  in  regard  to 
failures  in  Grades  VII  and  VIII.  In  the  compiling  of 
this  table,  it  was  considered  (1)  that  a  pupil  had  failed 
to  do  a  certain  type  if  he  did  not  get  at  least  one  example 
right  among  those  he  attempted,  (2)  that  a  pupil  did 
not  fail  if  he  used  the  correct  method  even  though  he 
did  not  get  at  least  one  right  answer.  The  table  is  to  be 
interpreted  as  follows.  In  Test  1,  3.1  per  cent  of  Grade 
VIII  failed  in  the  multiplication  of  an  integer  by  a  fraction 
and  2.6  per  cent  failed  in  the  multiplication  of  a  fraction 
by  an  integer.  In  Test  2,  17.9  per  cent  failed  in  the 
multiplication  of  a  mixed  number  by  an  integer  and 
30.6  per  cent  failed  in  the  multiplication  of  an  integer 
by  a  mixed  number  and  so  on. 

In  multiplying  a  mixed  number  by  a  mixed  number, 
Test  4,  two  forms  were  used  as  illustrated  below. 

(a)     Vertical  method :  (6)     Horizontal  method : 
32i  32i  X  69|  = 
69i  97       139  _  13483  _ 
3   ^    2     "      6      ~  ^"^^^^ 


XU6 

23 

288 
192 


2247J 


In  this  paper  whenever  the  example  was  done  similarly 
to  illustration  (a),  it  has  been  termed  the  vertical  method, 
and  when  (h)  the  mixed  numbers  were  reduced  to 
improper  fractions,  it  has  been  termed  the  horizontal 
method.  When  the  example  was  done  by  the  vertical 
method,  the  eighth  grade  failed  in  90.4  per  cent  of  the 
cases;  when  done  by  the  horizontal  method  the  same 
grade  failed  in  4.6  per  cent  of  the  cases. 

In  Test  6  the  pupils  were  required  to  divide  a  mixed 
number  by  an  integer  (Examples  1  and  2)  or  an  integer 
by  a  mixed  number  (Examples  3  and  4).  Two  possi- 
bilities of  doing  the   work  present  themselves.     First, 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  21 

pupils  may  reduce  the  mixed  numbers  to  improper 
fractions  and  follow  the  general  rule  for  division  of 
fractions. 

For  example  (a) :    ^^-o^       r       17035       _ 

ODYO^  -^  O   =  — -r-  5   = 


Second,  they  may,  when  the  fraction  is  in  the  dividend, 
do  the  example  by  either  short  or  long  division  as  it 
stands. 

For  example  (h) :      5678 J  -^  5 
5)56781 

1135  Rem.  =  3 J 

^3  •-      "  3  ^  5  "  3 
11351  Ans. 

When  the  mixed  number  is  in  the  divisor,  they  may 
place  the  example  on  the  paper  as  though  they  were  doing 
an  example  in  long  division,  multiply  both  dividend  and 
divisor  by  the  denominator  of  the  fraction  and  proceed 
as  in  long  division. 

For  example  (c):   2467 -- 8i  33)9868^        ^^^' 

66 
gl)2467  ^ 

4        4  297 


33) 9868  298 

297 

1 

These  two  processes  will  be  termed   (a)   process  of 
inversion  and  (b  and  c)  process  of  long  division. 

Diagnosis  of  Results  in  Each  Test 

TEST   1. 
Type  of  Examples  Used  in  Test  1. 

(1)     i  X  6      (2)     *  X  8      (3)     f  X  12       (4)     12  X  A 


22  SCHOOL   DOCUMENT   NO.  5. 

In  this  test  the  pupils  were  required  to  multiply  a 
fraction  by  an  integer  (Examples  1,  2  and  3)  or  an 
integer  by  a  fraction  (Example  4).  About  13  per  cent 
failed  in  the  seventh  grade  and  about  3  per  cent  failed 
in  the  eighth  grade  in  each  type.  In  such  a  simple  test 
the  chances  of  making  errors  are  limited;  so  they  fall 
very  largely  into  two  groups.  In  one  group  the  pupils 
find  the  answer  by  multiplying  the  integer  by  one  of  the 
terms  of  the  fraction  and  adding  the  other. 

For  example:  i  X  6  =  14     (6  X  1  +  8)  or 
i  X  6  =  49     (6X8  +  1) 

The  pupils  in  the  second  group  multiply  both  numer- 
ator and  denominator  by  the  integer. 

^  ,      7  ^  c       56  (8  X  7) 

Forexample:    -X8  =  -^g^gj 

Cancellation  gives  little  trouble  because  compara- 
tively few  pupils  use  this  method  of  shortening  the 
procedure.  In  some  cases  there  was  evidence  of  can- 
celling by  dividing  the  integer  by  the  numerator,  but 
these  cases  were  few.  There  seemed  to  be  a  mixture  of 
processes  in  the  minds  of  some  pupils  because  a  few 
inverted  one  or  the  other  of  the  factors.  . 

TEST  2. 
Type  of  Examples  Used  in  Test  2. 

(1)     246i     (2)     2731     (3)     275       (4)     456^      (5)     189 

5  5  8f  2  5i 


In  this  test  the  pupils  were  required  to  multiply  a 
mixed  number  by  an  integer  (Examples  1,  2  and  4)  or 
an  integer  by  a  mixed  number  (Examples  3  and  5). 
The  percentage  of  failure  for  the  first  type  for  Grade  VII 
was  34.3  per  cent  and  for  Grade  VIII,  17.9  per  cent. 
For  the  second  type  the  percentage  was  nearly  twice  as 
much,  being  53.4  per  cent  and  34.3  per  cent  respectively. 
This  great  difference  was  due  very  largely  to  the  con- 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  23 

struction  of  two  examples,  the  fourth,  456yX  2,  and  the 
thirteenth,  379i  X  3.  In  many  cases  pupils  had  the 
fourth  example  right  and  also  the  thirteenth,  if  they 
reached  this  example,  and  no  others.  In  these  two 
examples  the  multiplication  of  the  fraction  by  the 
integer  gives  the  fractional  part  of  the  product  without 
further  reduction. 

In  this  test  there  are  three  chief  sources  of  error. 

(a)  If  the  fraction  be  in  the  multiplicand,  the  multi- 
plication of  the  fraction  by  the  integer  in  the  multiplicand 
and  if  the  fraction  be  in  the  multiplier,  the  multiplication 
of  the  fraction  by  the  integer  in  the  multipher. 


For  example:  246^ 
5 

275 

81 

49i 
1230 

=  i  X  246 
=  5  X  246 

6  =  1X8 
2200  =  8  X  275 

1279i 

Ans. 

2206      Ans. 

(b)     Placing  of   the  second  partial  product  one  place 
3  the  left  of  the  first  partial  product. 

For  example:  275 

81 

i  X  275 
8  X  275 

206i  = 
2200      = 

» 

22206i     Ans. 

(c)     Multiplication  of  the  denominator  of  the  fraction 
by  the  integer  and  adding  the  numerator. 

For  example:  246^ 
5 


26  =  5  X  5  +  1 
1330 


1356 


The  kind  of  error  in  (a)  develops  because  it  is  not  clear 
in  the  minds  of  many  pupils  which  integer  is  to  be  mul- 
tiplied by  the  fraction.  If  the  fraction  be  in  the  mul- 
tiplier, the  integer  of  the  multiplier  is  multiplied. by  the 


24  SCHOOL  DOCUMENT   NO.  5. 

fraction  and  this  product  given  as  one  of  the  partial 
products.  It  is  not  strange  that  this  should  be  done  on 
account  of  the  drill  which  has  been  given  in  reduction 
of  mixed  numbers  to  improper  fractions.  In  other  words 
an  old  habit  is  at  work,  for  the  pupil  has  not  yet  appre- 
ciated that  it  is  not  the  same  thing  but  something  entirely 
different.  For  those  pupils  who  persist  in  doing  this 
work,  individual  attention  is  probably  the  only  method 
of  eradicating  the  error. 

The  kind  of  error  noted  in  (b)  is  due  probably  to  the 
same  cause,  viz.,  the  following  of  an  old  habit.  In 
multiplication  of  integers  the  pupil  was  taught  that  he 
must  place  the  second  partial  product  one  step  to  the  left 
of  the  first  partial  product.  When  the  process  in  frac- 
tions is  performed,  the  pupil  follows  the  same  habit 
unless  he  is  led  to  see  the  difference  and  a  large  amount 
of  practice  in  the  correct  method  of  doing  the  work  is  given. 
Some  pupils  will  need  more  of  this  practice  than  others 
before  the  new  habit  is  fixed. 

The  third  source  of  error  noted  in  (c)  develops  through 
the  multiplication  of  the  fraction  by  the  integer.  For 
example,  it  was  a  fairly  common  error  in  such  an  example 
as  2461  X  5  to  call  5  X  |  ==  26.  That  is,  apparently  the 
example  was  done  exactly  as  though  it  was  reduced  to 
an  improper  fraction  and  then  the  denominator,  5, 
thrown  away. 

Another  habit  is  probably  at  work  in  this  case  which 
is  not  generally  taken  into  consideration.  In  teaching 
multiplication  of  integers  emphasis  is  placed  upon  the 
fact  that  the  product  of  one  number  by  another  is  larger 
than  either  of  the  factors.  To  have  a  pupil  realize 
that  a  number  may  be  multiplied  by  another  such  that 
the  product  is  smaller  than  one  of  the  factors  and  that 
when  both  factors  are  fractions  the  product  is  smaller 
than  either  of  the  fractions,  means  that  the  pupil  must 
break  old  habits  and  form  new  ones.  The  ability  to  meet 
this  new  experience  and  use  it  means  a  large  amount  of 
drill  before  the  old  habit  can  be  modified  to  meet  the 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  25 

new  conditions.  Unless  this  drill  is  adequate  such 
errors  as  those  just  pointed  out  are  likely  to  occur. 

Emphasis  is  being  placed  upon  estimating  the  answer 
in  many  schools,  but  even  though  a  pupil  is  able  to  make 
an  approximate  estimate  of  the  product  in  these 
examples,  he  will  not  be  able  to  trace  the  error  until  he 
appreciates  the  possibilities  of  these  three  types  of 
error. 

Many  strange  methods  were  used  which  do  not  fall 
into  the  foregoing  groups.  The  following  examples  are 
illustrations  of  these: 


341 

71 

=  341  -T-  3 

7061 

7 

1131 

5651  =  8  X  706  +  3 

56i 

=  113  ^  2 

4942  =  7  X  706 

2387 

=  7  X  341 

10593   Ans. 

2556^    Ans. 

If  teachers  will  give  a  test  similar  to  this  one  to  their 
classes,  it  is  more  than  likely  that  some  of  these  strange 
ways  of  doing  the  work  will  manifest  themselves.  It 
should  be  clear  that  class  work  does  not  reach  these 
individuals  and  if  the  pupil  is  to  learn  the  correct  method 
it  is  only  through  individual  help. 

TEST  3. 
Type  of  Examples  Used  in  Test  3. 

(1)     ^  Xi     (2)     7i  X  f     (3)     5i  X  f     (4)     f  X  2f 

In  this  test  the  pupils  were  required  to  multiply  a 
mixed  number  by  a  fraction  (Examples  1,  2  and  3),  or 
a  fraction  by  a  mixed  number  (Example  4) .  The  per- 
centage of  failure  was  about  17  per  cent  for  Grade  VII 
and  8.7  per  cent  for  Grade  VIII  in  each  test. 

The  greatest  difficulties  in  this  test  are  shown  in  (1) 
the  reduction  of  the  mixed  number  to  an  improper 
fraction  and  (2)  in  the  process   of   cancellation.     The 


26  SCHOOL  DOCUMENT  NO.  5. 

first  type  of  failure  shows  itself  in  many  ways.  For 
example,  some  pupils  multiply  the  two  fractions  and 
add  the  integer. 

For  example:    4|  X  |  =  4/:^. 

Some  pupils  consider  the  integer  as  another  factor 
instead  of  a  part  of  one  of  the  factors. 

17  1        5  K^  o2       20     (5X2X2      20). 

For  example:    I  X  2f  =  -     J-^^^^  = -j^ 

Another  common  method  was  to  multiply  the  integer 
by  the  numerator  of  the  fraction  and  add  the  numerator 
of  the  fraction  which  is  a  part  of  the  mixed  number. 

For  example-   ^  X  3^  -  ^-^    (^  X  ^  +  ^  -  15). 
J^or  example.    ,  x  6^  -  ^^    ^    ^^^      -  ^^^^ 

There  were  many  other  improper  methods  of  finding 
the  product  of  a  mixed  number  and  a  fraction. 

In  cancellation  the  difficulty  came  in  canceling  before 
the  reduction  of  the  mixed  number  to  the  improper 
fraction  and  also  in  canceling  the  integer  of  the  mixed 
number.  Examples  of  this  type  of  error  seem  unneces- 
sary. 

TEST  4. 
Type  of  Examples  Used  in  Test  4. 
(1)     32i         (2)     S^         (3)     29f         (4)     25f         (5) 


79i 


8 
3  »'  2 


In  this  test  the  pupils  were  required  to  multiply  a 
mixed  number  by  a  mixed  number. 

There  were  78  per  cent  of  the  pupils  who  attempted 
to  do  the  work  vertically  and  22  per  cent  who  did  it 
horizontally.  Of  those  who  did  the  work  vertically  92 
per  cent  failed  to  do  the  work  correctly,  and  1  per  cent 
had  the  method  correct  but  made  errors  in  the  work, 
while  the  remaining  7  per  cent  had  the  correct  answer. 
Of  those  who  did  the  work  horizontally  7  per  cent  failed 
to  do  the  work  correctly,  45  per  cent  had  the  method 
correct  but  made  errors  in  the  work,  while  48  per  cent  had 
the  correct  answer.  The  two  methods  will  be  con- 
sidered separately. 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  27 

(a)  In  a  study  of  the  vertical  method  it  is  noticeable 
that  only  a  small  percentage  of  the  pupils  (8  per  cent) 
even  get  the  right  method.  There  were  three  common 
erroneous  ways  of  doing  the  work. 

1.  The  product  or  sum  of  the  fractions  added  to 
product  of  the  integers. 

For  example : 
o2-o  o2-o 


288    =9X32 

or 

288     =9X32 

192     =  6  X  32 

192       =  6  X  32 

i  =  4Xi 

1  =  4  +  1 

2208J  2208f 

By  far  the  largest  proportion  of  the  pupils  who  did 
the  work  vertically  found  their  answers  in  this  manner. 
2.     A  disregard  of  one  or  both  of  the  fractions. 


For  example :     32  J 

321 

69i 

69J 

16    =  i  X  32 

or      288    =  9  X  32 

288    =  9  X  32 

192      =  6  X  32 

192 

2224  2208 

3.     Process  correct  except  that  the  product  of  the  fractions 
is  omitted. 

For  example :    32 J 


691 

23  = 

JX69 

16  = 

iX32 

288  = 

9  X32 

192    = 

6X32 

2247 

(b )  The  chief  error  when  the  work  is  done  horizon- 
tally consists  of  inverting  one  or  both  of  the  fractions 
after  the  mixed  numbers  have  been  reduced  to  improper 
fractions. 


28  SCHOOL   DOCUMENT  NO.  5. 

It  is  not  the  intention  of  this  bulletin  to  point  out  the 
method  to  be  followed  in  multiplication  of  mixed  num- 
bers. There  are  three  facts,  however,  that  are  signifi- 
cant. 

1.  The  larger  proportion,  78  per  cent,  of  the  pupils 
did  their  work  vertically  either  through  habit,  or  because 
the  examples  were  placed  in  a  vertical  position  in  the 
test,  or  because  of  ignorance,  and  therefore  followed  the 
line  of  least  resistance. 

2.  Of  those  who  did  the  examples  vertically  and  used 
the  right  method,  only  a  small  percentage  made  errors. 

3.  There  was  a  large  percentage,  45  per  cent,  who  used 
the  correct  method  when  doing  the  work  horizontally 
but  made  errors  in  the  work.  It  is  not  fair,  however, 
to  draw  the  conclusion  that  the  better  method  is  the 
vertical  method,  because  of  the  high  percentage  of 
accuracy.  The  number  of  pupils  concerned  is  not  large 
enough  to  make  this  conclusive. 

In  multiplying  mixed  numbers  horizontally  the  pupil 
must  take  the  following  steps  to  do  the  work: 

Reduction  of  one  mixed  number  to  improper  fraction. 

1.  Integer  multiplied  by  the  denominator. 

2.  Add  the  product  to  the  numerator. 

3.  Write  the  improper  fraction. 

Reduction  of    second  mixed    number    to    improper 
fraction. 

4.  Same  as  1  (above)  for  second  fraction. 

5.  Same  as  2  (above)  for  second  fraction. 

6.  Same  as  3  (above)  fo^  second  fraction. 
Multiplication  of  improper  fractions. 

7.  Numerator  of  first  fraction  multiplied  by  numer- 

ator of  second  fraction. 

8.  Denominator   of    first   fraction   multiplied  by 

denominator  of  second  fraction. 

9.  Reduction  of  improper  fraction  to  a  mixed  num- 

ber to  find  the  answer  (7  -J-  8). 

Thus  in  doing  this  test  multiplication  and  division  of 
integers  play  an  important  part.     When  one  considers 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  29 

that  the  median  accuracy  in  multiplication  of  integers 
is  80  per  cent  and  the  median  accuracy  in  division  of 
integers  is  90  per  cent,  such  a  large  percentage  of  error 
as  shown  in  doing  this  type  of  examples  by  the  horizon- 
tal method  leads  one  to  suspect  that  there  must  be  some 
factor  present  which  is  not  being  considered  but  has  an 
important  influence  upon  the  result. 

The  error  may  occur  in  any  one  of  the  nine  steps  noted 
in  the  foregoing.  An  error  in  the  early  steps  may  result 
in  an  increased  error  in  the  answer.  Which  method 
should  be  used  in  multiplication  of  mixed  numbers 
depends  on  two  important  questions  to  neither  of  which 
do  we  have  an  answer. 

1.  By  which  method  is  the  pupil  most  likely  to  obtain 
the  correct  answer? 

2.  How  long  does  it  take  pupils  to  learn  the  method? 
With  sixth  grade  pupils  it  may  be  possible  that  the 

time  used  to  teach  the  vertical  method  would  be  out  of 
proportion  to  the  results  achieved  through  a  greater 
percentage  of  accuracy  providing  this  method  is  more 
likely  to  produce  accurate  results.  It  may  be  that  pupils 
of  sixth  grade  ability  are  not  mature  enough  to  learn 
the  vertical  method  without  an  unreasonable  expendi- 
ture of  time.  If  this  be  true,  the  teaching  of  this  process 
according  to  this  method  should  be  left  to  the  high  school 
classes.  Whatever  may  be  the  truth  in  the  case,  if  we  are 
going  to  teach  multiplication  of  mixed  numbers,  50  per 
cent  of  the  eighth  grade  class  should  obtain  a  higher 
accuracy  than  0. 


TEST 

5. 

Type  of  Examples 

Used 

in 

Tests 

3 

4 

-8      (2) 

9  -V-  f 

(3) 

6 

-1 

(1)     i  -  8      (2)     9  -V-  f      (3)     6  -  I      (4)     8  ^  f 

Tests  5,  6,  and  7  consisted  of  examples  in  division  of 
fractions.  In  Test  5  the  pupils  were  required  to  divide 
an  integer  by  a  fraction  (Examples  2,  3,  and  4)  or  a 
fraction  by  an  integer  (Example  1).  The  percentage 
of  error  was  very  large,  being  35.4  per  cent  for  the  first 


30  SCHOOL  DOCUMENT  NO.  5 

type  and  49.5  per  cent  for  the  second  type  in  Grade 
VII  and  25.3  per  cent  and  42.9  per  cent  for  Grade  VIII 
in  the  respective  types. 

The  chief  cause  for  failure  to  get  at  least  one  example 
right  in  such  a  large  percentage  of  cases  is  due  to  failure 
to  invert  the  divisor.  Either  the  pupil  does  not  invert 
the  divisor  or,  not  knowing  which  is  the  dividend  and 
which  the  divisor,  inverts  the  dividend.  A  notable  fact 
is  that  when  the  divisor  is  an  integer  the  chance  of 
failure  is  nearly  doubled.  Evidently  the  pupil  does  not 
know  the  possibilities  in  this  case. 

As  pointed  out  under  the  discussion  of  the  fourth  test, 
habit  plays  a  very  important  part.  The  particular  habit 
which  probably  influences  the  results  in  this  instance  is 
one  formed  in  work  with  integers.  Here  the  pupil 
learned  that  the  answer  in  division  must  be  smaller 
than  the  dividend.  There  certainly  comes  a  new  experi- 
ence into  the  hfe  of  the  pupil  when  he  sees  for  the  first 
time  a  division  example  in  which  the  answer  is  larger 
than  the  dividend.  Unless  it  is  made  very  clear  it  must 
be  a  difficult  thing  for  pupils  to  understand  how  one  can 
divide  4  by  J  and  get  an  answer  of  16.  To  get  such  a 
large  answer  seems  to  violate  all  their  previous  concep- 
tion of  the  meaning  of  division.  It  may  be  possible 
that  the  teaching  of  the  idea  of  partition  and  measuring 
as  pointed  out  in  the  earlier  part  of  the  bulletin  would 
be  a  material  help  in  conquering  this  difficulty. 

The  working  of  old  habits  may  then  be  a  partial 
explanation  of  the  cause  of  such  a  large  percentage  of 
error  in  such  simple  examples  as  those  given  in  Test  5. 

The  method  of  doing  the  examples  was  very  largely 
the  method  of  inversion.  There  were  a  few  cases  of 
the  longer  and  more  laborious  method  of  reduction  to 
a  common  denominator  and  then  dividing  one  numera- 
tor by  the  other.  These,  however,  were  isolated  cases 
and  need  only  be  mentioned  in  passing. 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  31 


(1)     56784 


TEST  6. 
Type  of  Examples  Used  in  Test  6. 

--  5         (2)     27891  ^  4         (3)     2467  ^  8^ 

(4)     6752  4-  12i 


Table  VIII  shows  the  results  in  Test  6.  The  table  is 
to  be  read  as  follows:  in  doing  the  type  mixed  number 
divided  by  an  integer,  18.3  per  cent  of  the  eighth  grade 
failed  to  do  correctly,  even  in  method,  any  of  the 
examples,  60.7  per  cent  did  the  work  correctly  in  at 
least  one  example,  and  21  per  cent  used  the  correct 
method  but  did  not  have  a  single  example  correct. 
The  rest  of  the  table  is  read  in  a  similar  way. 


TABLE  VI I L 
Showing  Results  Attained  in  Test  6. 

Division  of  Fractions. 


Mixed  Number  Divided  by 

Whole  Number  Divided  by 

Whole  Number. 

Mixed  Number. 

INVERSION. 

LONG  DIVISION. 

inversion. 

LONG  DIVISION. 

Grade, 

§ 

^ 

u 

J 

(0 

ij 

+3    O 

.jj   . 

-»^  2 

f^-e 

0  ii 

"S-3 
e^^ 

^■s 

Pi  S 

Jl 

el's 

cJ  t 

6& 

flif 

d  U 

' 

n^ 

6-^ 

6^ 

6-^ 

6^ 

c")!^ 

a^ 

6-^ 

6^ 

uf4 

^rt 

^•s 

Mrt 

u^i 

uK 

v" 

Ph 

(u 

PLH 

fin 

P4 

Pt 

pL. 

^1 

PLH 

Pk' 

Ph 

(U 

vni 

18.3 
30.7 

60.7 
40.3 

21 
29 

83.4 
87.2 

15.5 
10.3 

1.1 

2.5 

20.0 
40.5 

31.4 
15.4 

48.6 
44.1 

97.3 
99.2 

2.7 
.5 

0 

vn 

.3 

The  failure,  when  the  work  was  done  by  inversion, 
was  in  inverting  the  wrong  fraction  as  pointed  out  in 
analysis  of  Test  5.  When  the  work  was  done  by  long 
division,  the  failure  was  the  inability  to  dispose  of  the 
remainder.  The  large  percentage  of  failures  in  both 
types  when  done  by  long  division  does  not  necessarily 


32  SCHOOL  DOCUMENT   NO.  5. 

show  anything  because  there  was  considerable  evidence 
that  this  type  of  division  of  fractions  had  not  been 
taught  by  many  teachers.  It  is  indeed  possible  to  come 
to  the  same  conclusion  as  reached  in  the  analysis  of 
Test  4,  viz.,  if  this  type  is  going  to  be  taught  (and  it 
is  required  by  the  new  course  of  study)  it  should  be 
taught  effectively  enough  so  that  the  median  accuracy 
should  be  more  than  39  per  cent  in  the  eighth  grade. 

TEST  7. 
Type  of  Examples  Used  in  Test  7. 

(1)     f-i      (2)     3f-^i      (3)     5|-|      (4)     6f -^  f 

In  this  test  the  pupils  were  required  to  divide  a  frac- 
tion by  a  fraction  (Example  1)  or  a  mixed  number  by  a 
fraction  (Examples  2,  3  and  4).  In  the  eighth  grade 
about  20  per  cent  and  24  per  cent  failed  respectively  in 
each  type,  and  in  the  seventh  grade  39  per  cent  and  33 
per  cent  failed  respectively.  The  general  cause  of  failure 
was  the  same  as  in  Test  5,  viz.,  difficulty  with  inversion 
in  one  form  or  another. 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  33 


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34 


SCHOOL  DOCUMENT  NO.  5. 


Analysis  of  Results  in  Gxade  VI. 

Attention  has  already  been  called  to  the  fact  that 

the  pupils  of  the  sixth  grade  were  apparently  unable  to 

do  the  work  in  multiplication  and  division  of  fractions. 

It  ^ems  worth  while,  however,  to  call  attention  to  the 


FIGURE    1. 
Median  Scores  for  Grade  VI  in  Ten  Schools  in  Multiplication  of  Fractions. 

Median  Scores  for  Tests  5,  6,  and  7  are  all  0. 

0  /O  ZO  JO  40  JO  GO  70  ^0  'fo  /Oi 

1%. 


11 


30     \ 


37 


bO 


Sb 


b% 


^0 


4h 


bl 


Iv- 

--^ 

1 

1 

— 

/ 

" — .^_^ 

L,-^**^ 

T^ 

^:^^^ 

"-^^Z^ 

\ 

i 

S, 

^ 

>^ 





^ 

:z^^=^^^ 

zz:^^^ 

--^^S^^ 

, — ■ — ^ 

'"■^ 

\ 

' 



Test  I Test  Z 7cst3 flsr  ■^ 

great  variation  in  the  accuracy  of  the  various  schools 
in  the  work  of  multiph cation  and  division  of  fractions. 

Table  IX  shows  the  results  in  accuracy  in  the  ten 
schools  which  were  tested.  In  the  first  column  is  shown 
the  school  designated  by  a  number.  Under  each  test 
is  shown  the  median  or  the  50  percentile  score,   the 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  35 

75  percentile  score,  which  is  the  score  above  which  are 
25  per  cent  of  the  cases  and  below  which  are  75  per  cent 
of  the  cases,  and  also  the  number  of  pupils  who  obtain 
a  rank  of  100  per  cent  accuracy.  That  is,  the  first  line 
is  read  as  follows:  in  school  No.  12  the  pupils  in  Test  1 


FIGURE  2. 

Seventy=five  Percentile  Scores  for  Grade  VI  in  Ten  Schools  in  Multiplication 

and  Division  of  Fractions. 

Seventy-five  Percentile  Score  for  Test  6  is  0. 


/Z 


X7 


30 


S7 


bo 


SU 


(y^ 


40- 


^b 


hi 


10 


ZO 


50 


r^ 

^ 

T^^^ 

^^^ 

^:^ 

*-v^ 

"^ 

^ 

/ 

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H^ 

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— ^^  "^ 

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— 

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^ 

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^^ 

^^ 

T^sT  / Test  a, Tcard  - TesT  4 TcstS — -  TesT  7     ^ 

have  a  median  of  0.  The  75  percentile  point  is  at  20 
and  7  pupils  reach  100  per  cent.  In  Test  2  the  median 
and  75  percentile  is  0  and  there  are  none  who  reach 
100  per  cent.  The .  results  in  the  other  tests  are  in 
succeeding  columns.  The  record  of  the  other  schools 
is  read  in  a  similar  way.     Attention  is  called  to  the 


36  SCHOOL   DOCUMENT  NO.  5. 

scores    attained  by    Schools    30,    60,    62,  61,  in   con- 
trast with  Schools  12,  57,  40. 

Figure  1  and  Figure  2  show  the  data  of  Table  IX  in 
graphic  form.  Figure  1  shows  the  variation  in  the 
medians.  The  records  for  Tests  5,  6,  and  7  are  not 
given  because  they  are  all  0  and  would  be  drawn  over 
the  line  showing  the  results  for  Test  4.  Figure  2  shows 
the  variation  in  the  75  percentile  scores  for  the  ten 
schools.  The  record  for  Test  6  is  not  drawn  because 
all  the  schools  had  a  record  of  0  the  same  as  in  Test  4. 
These  two  graphs  emphasize  the  great  variation  in  the 
results  much  more  strongly  than  Table  IX. 

As  before  pointed  out,  it  was  not  expected  that  the 
results  of  Grade  VI  would  be  high  because  of  lack  of 
preparation  in  Grade  V,  but  neither  was  so  great  a 
variation  expected  among  the  different  schools.  This 
difference  in  variation  is  probably  due  to  a  difference  in 
procedure  in  the  different  schools. 

> 
IV.     Plan  of  Diagnosis  for  Teacher. 

As  a  result  of  this  study  of  common  fractions,  how 
may  a  teacher  effectively  check  the  work  in  her  grade? 
How  may  a  teacher  determine  and  keep  in  some  perma- 
nent form  a  record  of  the  pupils'  ability  in  common 
fractions?  The  following  form  of  record  (page  37)  has 
been  used  and  proved  to  be  an  effective  method  of 
doing  this  work.  This  sheet  may  be  duplicated  so  that 
each  pupil  may  have  a  copy.  It  was  planned  to  be  used 
as  follows : 

The  teacher  may  give  examples  in  addition  similar  to 
the  types  illustrated  in  the  sixth  grade  course  of  study, 
School  Document  No.  19,  1917,  and  if  the  pupils  get  the 
answers  right  the  type  may  be  checked  in  column  marked 
''R'^  if  wrong  it  may  be  checked  in  column  marked 
^'W^\  These  checks  in  the  '^ wrong''  column  should 
be  changed  as  fast  as  the  pupil  has  mastered  the 
type.  After  giving  a  series  of  lessons  covering  the 
various  types  and  recording  results,  the  teacher  has  a 
record  of  the  ability  of  each  individual  in  the  room 
showing  his  strength  and  weakness.    If  a  pupil  fails  in 


ACHIEVEMENT  OF  PUPILS  IN  COMMON  FRACTIONS.  37 


a  problem,  it  can  be  determined  immediately  by  refer- 
ring to  the  record  and  by  asking  a  few  judicious  questions, 
whether  the  difficulty  is  in  the  mechanics  of  the  prob- 
lem or  in  the  problem  itself,  or  both.  In  any  case  the 
teacher  can  easily  tell  to  what  extent  she  is  required  to 
give  help  to  the  pupil. 

Record  in  Common  Fractions  of 

Name School 

Age Room 

Grade 


R. 

W. 

R. 

W. 

Addition. 
Type    1    

Multiplication. 

Fraction  by  an  integer 

Integer  by  a  fraction 

Integer  by  a  mixed  number, 

Mixed  number  by  an  integer, 

Fraction  by  a  mixed  number. 

Mixed  number  by  a  fraction, 

Fraction  by  a  fraction 

Mixed  number  by  a  mixed 
number 

2 

3 

4 

5 

6 

7 

8     .            

9 

10 

11 

12      .    . 

13 

14 

Mixed  numbers 

Subtraction. 

("Without  "Borrowing.") 

Type  1 

• 

Division. 

1 
j 

Fraction  by  an  integer 

Integer  by  a  fraction 

Integer  by  a  mixed  number. 

Mixed  number  by  an  integer. 

Mixed  number  by  a  fraction. 

Fraction  by  a  fraction 

2 

3 

4 

5            .    . 

6 

7 

8 

9. .    .  . 

Fraction  from  mixed  number, 

Mixed    number    from    mixed 
number  (with  "borrowing"), 

Fraction  from  integer 

Fraction  from  mixed  number. 

Mixed  number  from  integer .  . 

Mixed   number   from   mixed 
number 

38  SCHOOL  DOCUMENT  NO.  5. 

Summary  and  Conclusions. 

1.  The  median  accuracy  in  all  but  the  simplest 
tests  in  multiplication  is  strikingly  low  in  some  schools 
and  high  in  others.  The  range  of  variation  in  the 
medians  of  the  ten  school  districts  tested  extends  from 
0  to  92  per  cent. 

2.  Analysis  of  results  in  multiplication  of  mixed 
numbers  and  division  of  a  mixed  number  by  an  integer 
and  of  an  integer  by  a  mixed  number  seems  to  indicate 
a  lack  of  drill  in  these  types  commensurate  with  their 
difficulty.  A  large  percentage  of  the  pupils  show  an 
utter  lack  of  knowledge  of  the  process. 

3.  In  the  tests  in  division  of  fractions,  the  chief 
source  of  error  is  in  the  apparent  inability  of  the  indi- 
vidual pupil  to  distinguish  between  the  dividend  and  the 
divisor.  This  results  in  an  inversion  of  either  dividend 
or  divisor  and  sometimes  both. 

4.  The  low  percentage  of  accuracy  in  Tests  4  and  6 
where  the  process  consists  of  a  number  of  steps  leads 
one  to  think  that  some  factors  are  influencing  the  results 
which  are  not  usually  considered  as  important. 

5.  The  ineffectiveness  of  the  instruction  as  indicated 
by  the  large  variation  within  the  class  is  again  shown  in 
these  tests  in  multiplication  and  division  of  fractions. 
Class  room  drills  tend  to  increase  the  difference  between 
the  individuals  of  the  class  by  increasing  the  abihty 
of  the  bright  pupil  and  not  reaching  the  slow  pupil.  The 
difficulty  of  the  individual  can  only  be  reached  by  indi- 
vidual instruction  whether  that  pupil  be  advanced  or 
retarded.  The  waste  through  nonpromotion,  poor 
attendance,  and  other  causes  may  be  eliminated.  It  is 
highly  important  that  we  find  out  the  reasons  for  failure 
through  the  analysis  of  results  and  apply  the  remedy 
needed  in  each  individual  case. 


ANNOUNCEMENT. 


Bulletins  published  bx.the  Department  are  distributed  by  the  Secretary 
of  the  School  CommittfeB,  who  will,  so  far  as  the  supply  on  hand  permits, 
fill  mail  applications;  f(|jtf  copies  when  such  requests  are  accompanied  by 
the  price  indicated. 


No.        I. 


No. 

II. 

No. 

III. 

No. 

IV. 

No. 

V. 

No. 

VI, 

No. 

VII 

No. 

VIII. 

No. 

IX. 

No.       X. 


No. 

XI. 

No. 

XIL 

No.  XIII. 

No. 

XIV. 

No. 

XV. 

Provisional  Minimum  and  Supplementary  Lists  of  Spelling. 

Words  for  Pupils  in  Grades  I  to  VIII. 
School  Document  No.  8.     1914.     Out  of  Print. 
Provisional  Miminum  Standards  in  Addition,    Subtraction 

Multiplication  and  Division  for  Pupils  in  Grades  IV  tr 

VIII. 
School  Document  No.  9.     1914.     Out  oj  Print. 
Educational  Standards  and  Educational  Measurement. 
School  Document  No.  10.     1914.     Out  of  Print. 
Spelling,   Determining  the  Degree  of  Difficult}^  of  Spelling 

Words. 
School  Document  No.  10.     1915.     Out  of  Print. 
Geography.    A  Report  on  a  Preliminary  Attempt  to  Measure 

Some  Educational  Results. 
School  Document  No.  14.     1915.     Out  of  Print. 
English.     Determining  a  Standard  in  Accurate  Copying. 
School  Document  No.  2.     1916.     Price,  7.  cents. 
Arithmetic.     Determining  the  Achievement  of  Pupils  in  the 

Addition  of  Fractions. 
School  Document  No.  3.     1916.     Price,  7  cents. 
Report  on  High  School  Organization  and  Expenditures,  1916. 
Printed  for  local  distribution  only. 
Penmanship.     Determining  the  Achievement  of  Elementary 

School  Graduates  in  Handwriting. 
School  Document  No.  6.     1916.     Price,  7  cents. 
Arithmetic.    The  Courtis  Standard  Tests  in  Boston,  1912-1915. 

An  Appraisal. 
School  Document  No.  15.     1916.     Price,  7  cents. 
Spelling.    The  Teaching  of  Spelling. 
School  Document  No.  17.     1916.     Price,  7  cents. 
Standards    in    Silent    Reading,    with    Suggestions    on    How 

Teachers  May  Test  Their  Pupils  in  Silent  Reading. 
School  Document  No.  18.     1916.     Price,  7  cents. 
Arithmetic.    The  Value  to  the  Teacher,  to  the  Principal  and 

to  the  Superintendent  of  Individual  and  Class  Records 

from  Standard  Tests. 
School  Document  No.  22.     1917.     Price,  7  cents. 
A  Plan  for  the  Promotion  of  Teachers  from  Merit  Lists. 
School  Document  No.  2.     1918.     Price,  7  cents. 
Arithmetic.      Determining    the    Achievement    of    Pupils    in 

Common  Fractions. 
School  Document  No.  5.     1918.     Price,  7  cents. 


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